3-5x+5-3x=12xx2-1

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· 12:55 06/08/2025

$\frac{3}{- 5 x + 5} - 3 x = \frac{12 x}{x^{2} - 1}$

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To solve the equation:

5x+53−x3=x2−112x
First, we need to find a common denominator for all terms. The denominators are 5x+5, x, and x2−1. We can factor the denominators: 5x+5=5(x+1) x is already in its simplest form. x2−1=(x−1)(x+1) The common denominator is 5x(x+1)(x−1). Before we proceed, we need to identify any values of x that would make the denominators zero. 5x+5=0⟹5(x+1)=0⟹x=−1 x=0 x2−1=0⟹(x−1)(x+1)=0⟹x=1 or x=−1 So, x=−1,0,1.

Now, we can multiply each term by the common denominator to eliminate the fractions:

$5x(x+1)(x-1) \left( \frac{3}{5(x+1)} - \frac{3}{x} \right) = 5x(x+1)(x-1) \left( \frac{12x}{(x-1)(x+1)} \right)$ $5x(x+1)(x-1) \frac{3}{5(x+1)} - 5x(x+1)(x-1) \frac{3}{x} = 5x(x+1)(x-1) \frac{12x}{(x-1)(x+1)}$ $3x(x-1) - 3 \cdot 5(x+1)(x-1) = 5x(12x)$ $3x^2 - 3x - 15(x^2 - 1) = 60x^2$ $3x^2 - 3x - 15x^2 + 15 = 60x^2$ −12x2−3x+15=60x2
Now, we can move all terms to one side to form a quadratic equation:

0=60x2+12x2+3x−15
0=72x2+3x−15
We can divide the entire equation by 3 to simplify it:

0=24x2+x−5
We can solve this quadratic equation using the quadratic formula, x=2a−b±b2−4ac , where a=24, b=1, and c=−5.

$x = \frac{-1 \pm \sqrt{1^2 - 4(24)(-5)}}{2(24)}$ $x = \frac{-1 \pm \sqrt{1 - 4(-120)}}{48}$ $x = \frac{-1 \pm \sqrt{1 + 480}}{48}$ $x = \frac{-1 \pm \sqrt{481}}{48}$ The two solutions are:

x1=48−1+481
x2=48−1−481
We need to check if these solutions are valid by ensuring they are not equal to −1,0, or 1. Since 481 is not an integer, neither of the solutions will be −1,0, or 1. So, both solutions are valid.

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